G/T
//Speck
BA //BT //BG.
Note that the arrowBA→BT can be identified with the compositionBA−∼→[X/T]→BT, where X = A\T, and the first morphism is an isomorphism by Corollary 1.16. The map H∗(BA) → H∗(X) induced by the projectionπ:X =A\T →BAis surjective. Indeed, using Künneth formula this reduces to the case whereT has dimension 1, which follows from Lemma 9.8 below. Note that πcan be identified with the compositionX →[X/T]'BA. Thus, by Proposition 9.6 applied to f: [X/T]→BT, the map
H∗(G/T)⊗H∗(BT)H∗(BA)→H∗(G/A)
is an isomorphism. We conclude by applying the fact that H∗(BT) → H∗(G/T) is surjective (Theorem 4.4).
Lemma 9.8. Let A be an elementary abelian `-group, and let X be a connected algebraic space endowed with anA-action such thatX is the maximal connected Galois étale cover of[X/A]whose group is an elementary abelian`-group. Then the homomorphism
(9.8.1) H1(BA,F`)→H1([X/A],F`) induced by the projection[X/A]→BAis an isomorphism.
Proof. For any connected Deligne-Mumford stack X, H1(X,F`) is canonically identified with Hom(π1(X),F`), and (9.8.1) is induced by the morphism
π1([X/A])→π1(BA)'A.
The assumption means thatA is the maximal elementary abelian`-quotient ofπ1([X/A]).
Proposition 9.9. Let X be an abelian variety over k, A =X[`] = Ker(`: X → X). Then the mapH∗(BA,F`)→H∗(X/A,F`)induced by the projectionX/A→BAis surjective.
Proof. We apply Lemma 9.8 to the morphism `:X →X, which identifies the target with X/A.
By Serre-Lang’s theorem [48, XI Théorème 2.1], this morphism is the maximal étale Galois cover ofX by an elementary abelian`-group. Thus H1(BA)→ H1(X/A) is an isomorphism. It then suffices to apply the fact thatH∗(X/A) is the exterior algebra ofH1(X/A).
10 Proof of the structure theorem
We proceed in several steps:
(1) We first prove Theorem 8.3 (b) when X is a Deligne-Mumford stack with finite inertia, and whose inertia groups are elementary abelian`-groups.
(2) We prove Theorem 8.3 (b) forX a quotient stack [X/G].
(3) For certain quotient stacks [X/G] we establish estimates for the powers ofF annihilating the kernel and the cokernel ofaG(X, K) (6.16.9).
(4) Using (3), we prove Theorem 8.3 (b) for Deligne-Mumford stacks with finite inertia.
(5) We prove Theorem 8.3 (a) and the first assertion of (b) for Artin stacks having a stratification by global quotients.
Construction 10.1. Let f:X → Y be a morphism of commutatively ringed topoi such that
`OY = 0, K∈D(X). The Leray spectral sequence off,
Eij2 =Hi(Y, Rjf∗K)⇒Hi+j(X, K),
gives rise to an edge homomorphism
(10.1.1) ef,K: H∗(X, K)→H0(Y, R∗f∗K),
which is a homomorphism ofF`-(pseudo-)algebras ifK∈D(X) is a (pseudo-)ring. The following crucial lemma is similar to Quillen’s result [36, Proposition 3.2].
Lemma 10.2.LetKbe a pseudo-ring inD(X). Assume thatc= cd(Y)<∞. Then(Keref,K)c+1= 0. Moreover, if K is commutative, then ef,K is a uniform F-isomorphism; more precisely, for b∈E20,∗, we haveb`n ∈Imef,K, wheren= max{c−1,0}.
Proof. We imitate the proof of [36, Proposition 3.2] (for the case of finite cohomological dimension).
We have E2ij = 0 for i > c. Consider the multiplicative structure on the spectral sequence (Example 3.15). As Keref,K =F1H∗(X, K), where F• denotes the filtration on the abutment, (Keref,K)c+1⊂Fc+1H∗(X, K) = 0. IfKis commutative andb∈Er0,∗, then the formuladr(b`) =
`b`−1dr(b) = 0 implies thatb`∈E0,∗r+1. Thus forb∈E0,∗2 ,b`n ∈E2+n0,∗ =E∞0,∗= Imef,K.
Construction 10.3. LetX be a Deligne-Mumford stack of finite presentation and finite inertia overk. By Keel-Mori’s theorem [28] (see [40, Theorem 6.12] for a generalization), there exists a coarse moduli space morphism
f:X →Y,
which is proper and quasi-finite. LetK∈Dcart+ (X,F`). Then Construction 10.1 and Lemma 10.2 apply tof andK with cd`(Y)≤2 dim(Y).
For any geometric pointtofY, consider the following diagram of Artin stacks with 2-cartesian squares:
Xt //
X(t) //
X
f
t //Y(t) //Y.
We have canonical isomorphisms
(10.3.1) (Rqf∗K)t
−∼→Hq(X(t), K)−∼→Hq(Xt, K),
the second one by the proper base change theorem (cf. [34, Theorem 9.14]). Therefore, if we let PY denote the category of geometric points of Y (Definition 7.1), the map
(10.3.2) H0(Y, Rqf∗K)→ lim
t∈P←−Y
Hq(X(t), K)−∼→ lim
t∈P←−Y
Hq(Xt, K),
is an isomorphism ifK∈D+c(X,F`), by Proposition 7.2. On the other hand, recall (8.1.1) that Rq(X, K) = lim
(x:S→X)∈C←− X
Hq(S, Kx) = Γ(CcX, Hq(K•)),
where Kx = x∗K and Hq(K•) denotes the presheaf on CX whose value at x is Hq(S, Kx). We define a categoryCf and functors
Cf ψ
ϕ
~~CX PY
as follows. The categoryCf is cofibered over PY by ψ. The fiber category of ψ at a geometric pointt→Y isCX(t). The pushout functorCX(t) → CX(z) for a morphism of geometric pointst→z is induced by the morphismX(t)→ X(z) (Remark 7.24). The functorsϕt:CX(t) → CX induced by the morphismsX(t)→ X define ϕ. Thus we have an inverse image map
(10.3.3) ϕ∗:Rq(X, K)→Γ(cCf, ϕ∗Hq(K•)).
By Lemma 8.6 we have horizontal map is the second isomorphism of (10.3.1). The mapι∗ is an isomorphism.
Proof. Assertion (a) follows from the definitions. For (b) it suffices to show thatϕis cofinal. Let τ: CX → Cf be the functor carrying an`-elementary pointx: [S/A]→ X, withsthe closed point of S, to the induced `-elementary point τ(x) : [S/A] → X(f(s)). Then we have ϕτ ' idCX, and a canonical natural transformationτ ϕ → idCf, carrying an object ξ: [S/A] → X(t) of Cf to the cocartesian morphismτ ϕ(ξ)→ξ in Cf above the morphism f(s)→t in PY. These exhibitτ as a left adjoint to ϕ. Therefore, by Lemma 10.5 below, ϕ is cofinal. For (c), it suffices again to show thatιis cofinal. LetX → X(t) be an étale atlas. Asf is quasi-finite, up to replacing X by a connected component, we may assume thatX is a strictly local scheme, finite over Y(t). Then X(t) '[X/G], where G = AutX(t)(x), xis the closed point of X. Let ξ: [S/A] → [X/G] be an
`-elementary point of [X/G]. The`-elementary point [x/A]→ Xt, endowed with the morphism in C[X/G] given by the diagram
Corollary 10.6. The assertion of Theorem 8.3 (b) holds if X is a Deligne-Mumford stack with finite inertia, whose inertia groups are elementary abelian`-groups. More precisely, ifc= cd`(Y), where Y is the coarse moduli space of X, then (KeraX,K)c+1 = 0 and for K commutative and
is an isomorphism. Indeed, by Proposition 10.4 (c) this will imply that the right vertical arrow in the diagram of Proposition 10.4 (a) is an isomorphism. As (10.3.2) is an isomorphism, ϕ∗ is an isomorphism (Proposition 10.4 (b)), andef,K = Leqf,K has nilpotent kernel and, if K is commutative, is an F-isomorphism (Lemma 10.2), it will follow that aX,K = L
aqX,K has the same properties with the same bounds for the exponents. Asf:X →Y is a coarse moduli space morphism, there exists a finite radicial extensiont0 →tand a geometric pointy0ofX abovet0 such that (Xt0)red ' BAutX(y0). Therefore we are reduced to showing that aX,K is an isomorphism for X =BAk, where A is an elementary abelian `-group. In this case, idBAk: BAk → BAk is a final object ofCBAk, so we can identify Rq(BAk, K) with Hq(BAk, K), and aqBA
k,K with the identity.
Corollary 10.7. SupposeX = [X/G]is a global quotient stack (Definition 8.2), where the action ofGon X satisfies the following two properties:
(a) The morphismγ: G×X→X×X,(g, x)7→(x, xg)is finite and unramified.
(b) All the inertia groups of Gare elementary abelian`-groups.
Then the assertions of Corollary 10.6 hold.
Proof. As γin (a) can be identified with the morphismX×[X/G]X →X×X, which is the pull-back of the diagonal morphism ∆[X/G]: [X/G] → [X/G]×[X/G] by X×X → [X/G]×[X/G], (a) implies that ∆[X/G] is finite and unramified. In particular, [X/G] is a Deligne-Mumford stack.
Moreover, as the inertia stack is the pull-back of ∆[X/G]by ∆[X/G], [X/G] has finite inertia. Taking (b) into account, we see that [X/G] satisfies the assumptions of 10.6, and therefore 8.3 (b) holds for [X/G].
Proposition 10.8. Theorem 8.3 (b) for global quotient stacks[X/G](Definition 8.2) follows from Theorem 8.3 (b) forGlinear.
Proof. Consider the system of subgroups Gi =L·A[m`i]·F ofG=L·A·F as in the proof of Theorem 4.6 (with Λ =F` and n =`), where m is the order of F. Note that every elementary abelian`-subgroup ofA·F is contained inA[m`]·F. As a consequence, every elementary abelian
`-subgroup of G is contained in G1, so that the restriction map R∗G(X, K) → RG∗
i(X, K) is an isomorphism fori≥1. Consider the commutative diagram
H∗([X/G], K) // aG2d(X, K) has nilpotent kernel and, ifKis commutative,aG4d(X, K) is a uniformF-surjection.
Proposition 10.9. Theorem 8.3 (b) holds for global quotient stacks of the form[X/G], where G is either a linear algebraic group, or an abelian variety.
Proof. Although by Proposition 10.8 it would suffice to treat the case whereGis linear, we prefer to treat both cases simultaneously, in order to later get better bounds for the power ofF annihilating the kernel and the cokernel of the mapaX,K(Corollary 10.10). We follow closely the arguments of Quillen for the proof of [36, Theorem 6.2]. IfGis linear, choose an embedding ofGinto a linear group L = GLn over k [13, Corollaire II.2.3.4], and a maximal torus T of L. If G is an abelian variety, letL=T =G. In both cases, denote byS the kernel of`:T →T, which is an elementary abelian`-group of ordern. We let Lact onF =S\L by right multiplication. Ifg∈L(k), and if {S}denotes the rational point ofF defined by the cosetS, the inertia group ofLat{S}gisg−1Sg.
Let us show that the diagonal action ofGonX×F (resp.X×F×F) satisfies assumptions (a) and (b) of Corollary 10.7. It suffices to show this forX×F. Consider the commutative square
L×L
∼ //L×L
F×L //F×F
where the horizontal morphisms are the morphismsγ: (x, g)7→(x, xg). As the vertical morphisms are finite and surjective, so is the lower horizontal morphism. Moreover, the latter is unramified.
Hence the morphismγ:F×G→F×F is finite and unramified. The same holds for the morphism γ: (X ×F)×G → (X ×F)×(X ×F), (x, y, g) 7→ (x, y, xg, yg), because it is the composite X×F×G→X×X×F×G→X×F×X×F, where the first morphism (x, y, g)7→(x, xg, y, g) is a closed immersion by the assumption thatX is separated and the second morphism (x, x0, y, g)7→
(x, y, x0, yg) is a base change of F ×G→ F ×F. So (a) is satisfied for X×F. Moreover, the inertia groups ofGonX×F are conjugate inLto subgroups ofS, so (b) is satisfied forX×F.
As in [36, 6.2], consider the following commutative diagram (10.9.1) in which the double horizontal arrows are defined by pr12 and pr13. By Corollary 10.7, aG(X × F,[pr1/idG]∗K) andaG(X×F×F,[pr1/idG]∗K) have nilpotent kernels and, ifKis commutative, are uniformF-surjections. To show thataG(X, K) has the same properties it thus suffices to show that the rows of (10.9.1) are exact.
First consider the lower row. The component of degree q is isomorphic by definition (6.16.8) to the projective limit over (A, A0, g)∈ AG(k)\of
(10.9.2) Γ(XA0, Rqπ∗r∗K)→Γ(XA0 ×FA0, Rqπ∗r∗[pr1/idG]∗K)
⇒Γ(XA0×FA0×FA0, Rqπ∗r∗[pr1/idG]∗K), where we have putr:= [1/cg]. In order to identify the second and third terms of (10.9.2), consider the following commutative diagram, where the middle and right squares are cartesian:
[X×F/G]
We have (by base change for the middle square)
pr∗1Rqπ∗(r∗K)−∼→Rqπ∗(id×pr1)∗r∗K'Rqπ∗r∗[pr1/idG]∗K.
By the Künneth formula for the right square, we have
Γ(XA0×FA0,pr∗1Rqπ∗r∗K)−∼→Γ(XA0, Rqπ∗r∗K)⊗Γ(FA0,F`).
Therefore we get a canonical isomorphism
Γ(XA0×FA0, Rqπ∗r∗[pr1/idG]∗K)−∼→Γ(XA0, Rqπ∗r∗K)⊗Γ(FA0,F`).
We have a similar identification forXA0×FA0×FA0, and these identifications produce an isomor-phism between (10.9.2) and the tensor product of Γ(XA0, Rqπ∗r∗K) with
(10.9.3) Γ(Speck,F`)→Γ(FA0,F`)⇒Γ(FA0×FA0,F`).
AsA0 is an elementary abelian `-subgroup of G, A0 is conjugate in Lto a subgroup of S, hence FA0 6=∅. It follows that (10.9.3), (10.9.2) and hence the lower row of (10.9.1) are exact.
In order to prove the exactness of the upper row of (10.9.1), consider the square of Artin stacks with representable morphisms,
(10.9.4) [(Y ×F)/G] //
[Y /G]
[F/L] //BL,
where Y is an algebraic space of finite presentation over k endowed with an action of G, the horizontal morphisms are induced by projection fromF and the vertical morphisms are induced by the embeddingG→L. The square is 2-cartesian by Proposition 1.11 andBS '[(S\L)/L] = [F/L]. By Propositions 9.6, 9.7 and 9.9, H∗([F/L]) is a finitely generated free H∗(BL)-module and the homomorphism
H∗([Y /G], K)⊗H∗(BL)H∗([F/L])→H∗([Y ×F/G],[pr1/idG]∗K)
defined by (10.9.4) is an isomorphism. Applying the above toY =X andY =X×F, we obtain an identification of the upper row of (10.9.1) with the sequence
H∗([Y /G], K)→H∗([Y /G], K)⊗H∗(BL)H∗([F/L])
⇒H∗([Y /G], K)⊗H∗(BL)H∗([F/L])⊗H∗(BL)H∗([F/L]), which is exact by the usual argument of faithfully flat descent.
Corollary 10.10. Let X = [X/G] be a global quotient stack, and assume that either (a) G is embedded inL= GLn,n≥1, or (b)Gis an abelian variety. LetK∈Dc+([X/G],F`)be a pseudo-ring. Letd= dimX. In case (a), let e= dimL/G,f = 2 dimL−dimG. In case (b), lete= 0, f = dimG. Then
(i) (KeraG(X, K))m= 0, wherem= 2d+ 2e+ 1,
(ii) for Kcommutative andy∈R∗G(X, K), we havey`N ∈ImaG(X, K)forN ≥max{2d+ 2e− 1,0}+ log`(2d+ 2f+ 1).
Proof. As in the proof of Proposition 10.9, letF =S\L. We have cd`((X×F)/G)≤2 dim((X× F)/G) = 2d+ 2e. As all inertia groups of G acting on X ×F are elementary abelian `-groups, by Corollary 10.7 we have (KeraG(X ×F,pr∗1K))m = 0, hence (i) by (10.9.1). For (ii), set aG(X, K) =a0,aG(X×F,pr∗1K) =a1,aG(X×F×F,pr∗1K) =a2. Denote byu0:H∗([X/G], K)→ H∗([X×F/G],[pr1/idG]∗K) (resp. v0:R∗G(X, K)→R∗G(X×F,[pr1/idG]∗K)) the left horizontal map in (10.9.1), andu1=d0−d1: H∗([X×F/G],[pr1/idG]∗K)→H∗([X×F×F/G],[pr1/idG]∗K) (resp.v1 =d0−d1:R∗G(X×F,[pr1/idG]∗K)→R∗G(X×F ×F,pr∗1K)), the map deduced from the double map (d0, d1) in (10.9.1). Asd0 andd1 are compatible with raising to the `-th power, so isu1 (resp.v1). LetN1 = max{2d+ 2e−1,0}. By Corollary 10.7 we have v0(y)`N1 =a1(x1) for somex1∈H∗([X×F/G],[pr1/idG]∗K). By (10.9.1) we havea2u1(x1) =v1a1(x1) = 0. Leth be the least integer≥log`(2d+ 2f+ 1). As above we have cd`((X×F×F)/G)≤2d+ 2f, so by Corollary 10.7 we getu1(x1)`h = 0, hence by (10.9.1)x`1h =u0(x0) for somex0∈H∗([X/G], K), and finallyy`N1 +h =a0(x0).
Remark 10.11.
(a) If in case (a) of Corollary 10.10, we assume moreover thatX is affine, then cd`((X×F)/G)≤ d+eand cd`((X×F×F)/G)≤d+f by the affine Lefschetz theorem [50, XIV Corollaire 3.2]. Thus in this case (i) holds form=d+e+ 1 and (ii) holds forN ≥max{d+e−1,0}+ log`(d+f+ 1).
(b) Let f: Y → X be a finite étale morphism of Artin stacks of constant degree d. As the compositeH∗(X, K) f
∗
−→H∗(Y, f∗K)−−−→trf,K H∗(X, K) is multiplication by d,f∗ is injective if d is prime to `. Thus, in this case, if KeraY,f∗K is a nilpotent ideal, then KeraX,K is a nilpotent ideal with the same bound for the exponent. This applies in particular to the morphism [X/H]→[X/G], whereH < Gis an open subgroup of index prime to`.
Proposition 10.12. Theorem 8.3 (b) holds if X is a Deligne-Mumford stack of finite inertia.
More precisely, ifc = cd`(Y), where Y is the coarse moduli space ofX, and if r (resp. s) is the maximal number of elements of the inertia groups (resp.`-Sylow subgroups of the inertia groups) of X, then (Kera(X, K))(c+1)((s−1)2+1)= 0, and for K commutative andb∈R∗(X, K), we have b`N ∈Ima(X, K)forN≥max{c−1,0}+max{r2−2r,0}+dlog`(2(r−1)2+1)e+dlog`((s−1)2+1)e.
Heredxefor a real number xdenotes the least integer≥x.
Proof. Consider the coarse moduli space morphismf:X →Y. For every geometric point tofY, there exists a finite radicial extension t0 → t and a geometric point y0 of X above t0 such that (Xt0)red 'BAutX(y0). Note that for any fieldE, a finite group Gof order m can be embedded into GLm(E), given for example by the regular representation E[G] of G. Moreover, if m 6=
2 or the characteristic of E is not 2, then G can be embedded into GLm−1(E), because the subrepresentation ofE[G] generated byg−h, whereg, h∈G, is faithful. Thus, by Remark 10.11, the mapaXt,K in Proposition 10.4 (c) satisfies (KeraXt,K)(s−1)2+1 = 0, and, forK commutative, aXt,K is a uniformF-surjection for all geometric pointst→Y with bound for the exponent given by max{r2−2r,0}+dlog`(2(r−1)2+ 1)e, independent oft. Thus (Ker lim
←−t∈PY
aXt,K)(s−1)2+1= 0, and Lemma 10.13 below implies that lim
←−t∈PY
aXt,K is a uniformF-surjection, with bound for the exponent given by max{r2−2r,0}+dlog`(2(r−1)2+ 1)e+dlog`((s−1)2+ 1)e. Hence, by Lemma 10.2 and Proposition 10.4,aX,K has the stated properties.
Lemma 10.13. Let C be a category, and let u: R → S be a homomorphism of pseudo-rings in GrVecC. If u is a uniform F-injection (resp. uniform F-isomorphism) (Definition 6.10), then lim←−Cuis also a uniform F-injection (resp. uniform F-isomorphism). More precisely, if m≥0 is an integer such that for every objectiof C and everya∈Kerui,am= 0(resp. and if n≥0is an integer such that for every objectiofCand everyb∈Si,b`n ∈Imui), then for everyx∈Ker lim←−Cu, xm= 0(resp. for every y∈lim←−CS and every integerN ≥n+ log`(m),yN ∈Im lim←−Cu).
Proof. Letx= (xi) be an element in the kernel of lim←−Cu. Sincexi is in Kerui,xm= (xmi ) = 0.
Assume now that u is a uniform F-isomorphism with bounds for the exponents given by m andn, and lety = (yi) be an element of lim←−CS. For every object iof C, take ai in Ri such that
In order to deal with the general case, we need the following lemma.
Lemma 10.14. Let u: R → S be a homomorphism of pseudo-rings in GrVecC endowed with a splitting (Definition 3.2). Then(Keru)R= 0. In particular,(Keru)2= 0.
Proof. Leta∈Keru, b∈R. Sinceu(a) = 0,ab=u(a)b= 0.
Proposition 10.15. The first assertion of Theorem 8.3 (b) holds.
Proof. Ifi:Y → X is a closed immersion,j:U → X is the complement, then the following diagram of graded rings commutes:
is induced by Ri!K → i∗K, hence has square-zero kernel by Lemma 10.14. Thus (Keru)2 = 0.
It follows that if bothaY,Ri!K andaU,j∗K have nilpotent kernels, thenaX,K has nilpotent kernel.
Using this, we reduce by induction to the global quotient case. In this case, the assertion follows from Propositions 10.8 and 10.9.
This finishes the proof of the structure theorem (Theorem 8.3 (b)).
Lemma 10.16. Let C be a category having finitely many isomorphism classes of objects. LetA be the category whose objects are the elementary abelian `-groups and whose morphisms are the monomorphisms. Let F: C → A be a functor. Let F be the presheaf of F`-algebras on A given byF(A) = S(A∨). Let G be a presheaf of F∗F-modules onC. Assume that, for every object xof C, G(x)is a finitely generated F(F(x))-module. ThenR= lim
←−x∈CF(F(x))is a finitely generated F`-algebra andS = lim
←−x∈CG(x) is a finitely generatedR-module.
Proof. We may assume that C has finitely many objects. For any monomorphism u: A →B of elementary abelian `-groups, F(u) :F(B) → F(A) carries S(B∨)GL(B) into S(A∨)GL(A). Thus A7→ E(A) = S(A∨)GL(A)⊂ F(A) defines a subpresheafEofF`-algebras ofF. As GL(A) is a finite group, by [48, V Corollaire 1.5]F(A) is finite overE(A) andE(A) is a finitely generatedF`-algebra.
For givenA and B, since GL(B) acts transitively on the set of monomorphisms u:A →B, the map S(B∨)GL(B) → S(A∨), restriction of F(u), does not depend on u. Thus E(u) only depends onA andB. Therefore, via the functor rk :A →NcarryingA to its rank,E factorizes through a presheafRon the totally ordered setN: we have a 2-commutative diagram
C F // is finite overE(B). By Lemma 10.19 below, for eachxinC,E(F(x)) is finite over
Q= lim
←−y∈C
E(F(y))'lim
←−y∈C
R(f(y)).
The rest of the proof is similar to the proof of the last assertion ofloc. cit. AsChas finitely many objects, there exists a finitely generatedF`-subalgebraQ0ofQsuch that, for eachxinC,E(F(x)) is integral, hence finite over Q0. Note that R is a Q-submodule, a fortiori a Q0-submodule, of Q
x∈CF(F(x)). For eachxin C, F(F(x)) is finite over E(F(x)), hence finite overQ0. It follows thatQ
x∈CF(F(x)) is finite overQ0. AsQ0is a noetherian ring,Ris finite overQ0, hence a finitely generated F`-algebra. Similarly, S is a finitely generated Q0-module, hence a finitely generated R-module. Note thatQis also finite overQ0, hence a finitely generatedF`-algebra, though we do not need this fact.
The first step of the proof of Lemma 10.19 consists of simplifying the limitQusing cofinality.
Among the functors
f1, f2, and f3 are cofinal, while f4 is not cofinal. It turns out that after making contractions of typesf1,f2, andf3, we obtain a rooted forest, of which the source off4 is a prototype.
For convenience we adopt the following order-theoretic definitions. We define a rooted forest to be a partially ordered setP such that P≤x={y ∈ P |y ≤x} is a finite chain for all x∈ P.
We define arooted tree to be a nonempty connected rooted forest. Let P be a rooted tree. For x, y∈ P, we say thaty is achildofxifx < yand there exists no z∈ P such thatx < z < y. By the connectedness ofP,m(x) = minP≤xis independent ofx∈ P, henceP has a least elementr, equal tom(x) for allx. We callrtheroot ofP.
Remark 10.17. Although we do not need it, let us recall the comparison with graph-theoretic definitions. A graph-theoretic rooted tree T is a connected acyclic (undirected) graph with one vertex designated as the root [41, page 30]. For a graph-theoretic rooted tree T, we let V(T) denote the set of vertices ofT equipped with the tree-order, withx≤y if and only if the unique path from the rootr to y passes through x. For any x∈ V(T), V(T)≤x consists of vertices on the path fromr to x, so that V(T)≤x is a finite chain. ThusV(T) is a rooted tree. Conversely, for any rooted treeP, we construct a graph-theoretic rooted tree Γ(P) as follows. LetG be the graph whose set of vertices isP and such that two verticesxandy are adjacent if and only ifyis a child ofxor xis a child ofy. Note that eachx≤x0 in P can be decomposed into a sequence x=x0 < x1 <· · · < xn =x0, n ≥ 0, each xi+1 being a child of xi, which defines a path from xtox0 in G. Thus the connectedness of P implies the connectedness of G. If G admits a cycle, then there existsy ∈ P that is a child of distinct elements xand x0 of P, which contradicts the assumption thatP≤y is a chain. Let rbe the root ofP. Then Γ(P) = (G, r) is a graph-theoretic rooted tree. We haveP =V(Γ(P)) andT = Γ(V(T)).
The next lemma is probably standard but we could not find an adequate reference.
Lemma 10.18. Let C be a category and let f:C → N be a functor. Let P be the set of full subcategories of C that are connected components of f−1(N≥n) for some n ∈ N. Order P by inverse inclusion: for elements S and T of P, we write S ≤T if S ⊃ T. Let ψ:C → P be the functor carrying an object x to the connected componentψ(x) of f−1(N≥f(x)) containingx, and letφ:P →Nbe the functor carryingS tominf(S). Then:
(a) f =φψ.
(b) ψ:C → P is cofinal (Definition 6.1) andφ:P →Nis strictly increasing.
(c) P is a rooted forest. Moreover, ifC has finitely many isomorphism classes of objects, thenP is a finite set.
Proof. (a) Let xbe an object of C. Asx∈ψ(x),φ(ψ(x)) = minf(ψ(x))≤f(x). Conversely, as ψ(x)⊂f−1(N≥f(x)),f(ψ(x))⊂N≥f(x), so thatφ(ψ(x))≥f(x). Thusφ(ψ(x)) =f(x).
(b) LetS∈ P. Note thatS is a connected component off−1(N≥φ(S)). By definition, (S↓ψ) is the category of pairs (x, S ≤ψ(x)). Note thatS ⊃ψ(x) implies that xis inS. Conversely, for xinS,S is a connected component off−1(N≥n) for n≤f(x), henceS⊃ψ(x). Thus (S↓ψ) can be identified withS, hence is connected. This shows that ψis cofinal. Now letS < T be elements ofP. We haveφ(S)≤φ(T). Ifφ(S) =φ(T) =n, thenS andT are both connected components off−1(N≥n), which contradicts with the assumptionS)T. Thusφ(S)< φ(T).
(c) LetS ∈ P. Let T, T0 ∈ P≤S. ThenT (resp.T0) is a connected components of f−1(N≥n) (resp.f−1(N≥n0)), andT andT0 both containS. Thus T ⊃T0 ifn ≤n0 and T ⊂T0 ifn≥n0. Therefore,P≤S is a chain. Asφis strictly increasing,φinduces an injectionP≤S →N≤φ(S), hence P≤S is a finite set. Therefore,P is a rooted forest. Note that forS∈ PandxinS, every objecty ofC isomorphic toxis also inS. Thus, ifChas finitely many isomorphism classes of objects, then P is a finite set.
Lemma 10.19. Let C be a category having finitely many isomorphism classes of objects and let f:C →Nbe a functor. LetRbe a presheaf of commutative rings onNsuch that, for each m≤n, R(m)is finite over R(n). LetQ= lim
←−x∈CR(f(x)). Then:
(a) For each objectxofC,R(f(x))is finite overQ.
(b) For each connected componentS ofC and eachr inS satisfying f(r) = minf(S), we have Im(Q→ R(f(r))) = Im(R(maxf(S))→ R(f(r))).
Proof. By Lemma 10.18, we may assume that Cis a finite rooted tree with root r. We prove this case by induction on #C. LetB⊂ Cbe the set of children ofr. For eachc∈B,C≥cis a rooted tree with rootcandQis the fiber product overR(f(r)) of the ringsQc= lim
←−x∈C≥c
R(f(x)) forc∈B.
IfB is empty, then C={r}and the assertions are trivial. IfB={c}, thenQ'Qc and it suffices to apply the induction hypothesis toQc. Assume #B >1. Let n= maxf(C), nc= maxf(C≥c), and let c0 ∈ B be such that nc0 = minc∈Bnc. The complement C0 of C≥c0 in C is a rooted tree with rootr, andQis the fiber product over R(f(r)) of the rings Qc0 andQ0 = lim
←−x∈C0R(f(x)).
By the induction hypothesis,A = Im(Qc0 → R(f(r))) = Im(R(nc0) → R(f(r))) and Im(Q0 → R(f(r))) = Im(R(n)→ R(f(r))), so that we have a cartesian square of commutative rings
Q α
0 //
β0
Q0
β
Qc0
α //A.
As α is surjective, we have Ker(α0) ' Ker(α) and α0 is surjective (cf. [17, Lemme 1.3]), which implies (b). Moreover, asβ is finite, β0 is finite. Indeed, if A=P
iaiβ(Q0), then for liftingsa0i ofai,Qc0 =P
ia0iβ0(Q). The assertion (a) then follows from the induction hypothesis applied to Qc0 andQ0.
Proof of Theorem 8.3 (a). Let (ji:Xi → X)i be a finite stratification ofX by locally closed sub-stacks. The system of functors (CXi → CX)i is essentially surjective. Thus the map
R∗(X, K)→Y
i
R∗(Xi, ji∗K)
is an injection. Thus, for the first assertion of Theorem 8.3 (a), we may assume thatX is a global quotient stack, in which case the assertion follows from Theorem 6.17 (a) and Proposition 8.7.
Let Hq(K•) denote the presheaf on CX whose value at x: S → X is Hq(S, Kx), where Kx = x∗K, so that Rq(X, K) = lim
←−CXHq(K•). Let N be the set of morphisms f in CX such that (H∗(F`•))(f) and (H∗(K•))(f) are isomorphisms. By Lemma 7.3, lim
←−CX Hq(K•) ' lim←−N−1CX Hq(K•) and similarly for H∗(F`•). We claim that N−1CX has finitely many isomor-phism classes of objects. Then lim←−N−1CX commutes with direct sums, and, by Lemma 10.16, R∗(X,F`) and R∗(X, K) are finitely generated R-modules for a finitely-generated F`-algebra R, hence the second assertion of Theorem 8.3 (a). Using again the fact that the system of functors (CXi → CX)i is essentially surjective, we may assume in the above claim that X = [X/G] is a global quotient. Consider the diagram (8.5.2). Note that the functorAG(k)\ → EG(π0) induces a bijection between the sets of isomorphism classes of objects, andEG(π0) is essentially finite by Lemma 6.20 (a), thusAG(k)\has finitely many isomorphism classes of objects. Moreover, as Eis essentially surjective, it suffices to show that, for every object (A, A0, g) ofAG(k)\, the category M−1PXA0 has finitely many isomorphism classes of objects. HereM is the set of morphismsf in PXA0 such that (E(A,A∗ 0,g)H∗(K•))(f) is an isomorphism. Let (Xi) be a finite stratification ofXA0 into locally closed subschemes such thatK|Xi has locally constant cohomology sheaves. For a giveni, all objects in the image ofPXi →M−1PXA0 are isomorphic. Moreover, the system of func-tors (PXi →PXA0)i is essentially surjective. Therefore,M−1PXA0 has finitely many isomorphism classes of objects.